Spectral Image Processing

ABSTRACT

A substantially rectangular spectral representation is synthesized, which is adapted to produce either (a) image capture device sensor outputs if applied to an image capture device or (b) color values if applied to corresponding analysis functions. Spectral expansion, which can be used in various image processing methods, is achieved with the synthesized spectral representation.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of priority to U.S. Provisional Patent Application Ser. No. 61/581,051 filed on Dec. 28, 2011, U.S. Provisional Patent Application Ser. No. 61/581,048 filed on Dec. 28, 2011, and U.S. Provisional Patent Application Ser. No. 61/733,551 filed on Dec. 5, 2012, all hereby incorporated by reference in their entirety.

TECHNICAL FIELD OF THE INVENTION

The present disclosure relates to color processing for digital images. More particularly, an embodiment of the present invention relates to processing methods for transforming between tristimulus and spectral representations of digital images and for processing such spectral representations.

BACKGROUND OF THE INVENTION

As used herein, the phrases “spectral synthesis” and “spectral synthesis for image capture device processing” may relate to processing methods that may be performed or computed to achieve accurate color output, e.g., from image capture devices. Tristimulus color processing models, such as RGB (red, green, blue), are commonplace. While RGB and other tristimulus models suffice for color identification, matching, and classification, such models may be inherently limited in relation to color processing. By its nature, light comprises a spectrum of electromagnetic energy, which generally cannot be represented completely by, for instance, a red, a green, and a blue color value. With RGB based information as well as tristimulus values corresponding to cone cells receptive to short, medium, and long wavelength light (e.g., blue, green, and red), the human visual system (HVS) attempts to infer an original, natural stimulus.

Multi-spectral systems typically capture, process, and display multi-spectral images. Multi-spectral cameras for example may output more than three channels. Output channels can be rendered with a multi-primary printer or display. Some multi-spectral systems are designed to render a print output with a reflectance spectrum that is nearly identical to a reflectance spectrum of an original object. Multi-spectral representations of images generally fall into two classes. The more common class measures intensity or reflectance over smaller intervals in wavelength, which generally necessitates use of more than three channels (e.g., more than channels R, G, and B) (see reference [1], incorporated herein by reference in its entirety). The less common class uses the Wyszecki hypothesis (see reference [2], incorporated herein by reference in its entirety) which characterizes reflectance spectra as being comprised of two components, a fundamental component which captures a perceptually relevant tristimulus representation plus a residual component which represents the gross features of the overall reflectance spectrum. Wyszecki labeled this residual component the metameric black. An example of this second class is the LabPQR color space. In the LabPQR representation, the tristimulus portion is the Lab color space while PQR represents the residual. For emissive rendering and presentation of images using electronic displays, reflectance spectra identity is not crucial.

A picture produced by a camera or other image capture device is generally not quite the same as what would be perceived by human eyes.

Processing inside an image capture device generally involves a 3×3 matrix that transforms sensor outputs into a color space of an output image. Results of applying this matrix transformation generally do not reproduce what would be perceived by human eyes unless spectral sensitivities of the image capture device's sensors can be represented as a linear combination of color matching functions. In many cases, magnitude of these errors in the results is not inconsequential.

Existing DSLR (digital single-lens reflex) cameras, for instance, may have a knob to select a different 3×3 matrix for different types of scenes (e.g., night, sports, cloudy, portrait, etc.). However, in practice, getting the color right in general and also, for instance, for certain memory colors, such as face (skin) tones, can be problematic.

BRIEF DESCRIPTION OF DRAWINGS

The accompanying drawings, which are incorporated into and constitute a part of this specification, illustrate one or more embodiments of the present disclosure and, together with the description of example embodiments, serve to explain the principles and implementations of the disclosure.

FIG. 1A depicts a spectral image processing method in accordance with an embodiment of the present disclosure.

FIG. 1B depicts a spectral expansion module.

FIG. 1C depicts a manipulation of image capture device color sensor outputs in accordance with an embodiment of the present disclosure.

FIG. 2 depicts (x, y) chromaticity space divided into rectangular band-pass and band-gap spectra.

FIG. 3 depicts three-parameter and four-parameter rectangular spectral representations.

FIG. 4 depicts colors represented in both a rectangular spectrum in the) domain as well as in chromaticity space.

FIGS. 5A-5C depict a circular representation of the) domain.

FIG. 6A depicts a spectral synthesis module in accordance with the present disclosure.

FIG. 6B depicts a wavelength determination module in accordance with the present disclosure.

FIG. 7 depicts a rectangular representation of RGB (red, green, blue) basis colors, white, magenta, and a RYGCBV (red, yellow, green, cyan, blue, violet) filter bank.

FIG. 8 depicts a sigmoid mapping curve relating input and output color intensities.

FIG. 9 depicts a digital camera processing method.

FIG. 10 depicts a simplified digital camera processing method.

FIG. 11 depicts output colors from a camera and actual colors for comparison.

FIG. 12 depicts CIE (International Commission on Illumination) 1931 color matching functions.

FIG. 13 depicts camera spectral sensitivities for a digital camera.

FIG. 14A depicts a method for synthesizing correct color outputs based on image capture device sensor outputs in accordance with an embodiment of the present disclosure.

FIGS. 14B-14C depict examples of substantially rectangular spectral representations.

FIGS. 15A-15D depict an image capture device processing method, in accordance with an embodiment of the present disclosure, using synthesized substantially rectangular spectra.

FIG. 16 depicts an example of image capture device processing that is equivalent to a mapping from captured RGB (red, green, blue) values to correct color XYZ (tristimulus values corresponding to CIE 1931 color space) values.

FIG. 17 depicts a method for determining image capture device gamut in accordance with an embodiment of the present disclosure.

FIG. 18 depicts band-pass and band-gap spectra in both linear and circular wavelength domains.

FIG. 19 depicts the behavior near the boundary between band-pass and band-gap spectra.

FIGS. 20A and 20B depict colors for rectangular spectra represented in a [λ_(↑),λ_(↓)] plane.

DESCRIPTION OF EXAMPLE EMBODIMENTS

In an example embodiment of the disclosure, a method for synthesizing a substantially rectangular spectral representation based on input color values is presented, the input color values being based on an input spectrum of an image, the method comprising: providing the input color values, wherein each input color value is associated with a corresponding analysis function; determining a first wavelength and a second wavelength of the substantially rectangular spectral representation based on the input color values; and computing a scale factor based on the first wavelength, the second wavelength, and any one of the input color values and its corresponding analysis function to synthesize the substantially rectangular spectral representation based on the input color values, wherein: the synthesized substantially rectangular spectral representation is adapted to produce the input color values if the synthesized substantially rectangular spectral representation were applied to the analysis functions corresponding to the input color values, and the first wavelength comprises a wavelength where the substantially rectangular spectral representation transitions from a lower value to the scale factor and the second wavelength comprises a wavelength where the substantially rectangular spectral representation transitions from the scale factor to the lower value.

In an example embodiment of the disclosure, a system that is configured to synthesize a substantially rectangular spectral representation based on input color values is presented, the input color values being based on an input spectrum of an image, wherein each input color value is associated with a corresponding analysis function, the system comprising: a wavelength determination module that is configured to determine a first wavelength and a second wavelength of the substantially rectangular spectral representation based on the input color values; and a scale factor computation module that is configured to compute a scale factor based on the first wavelength, the second wavelength, and any one of the input color values and its corresponding analysis function to synthesize the substantially rectangular spectral representation based on the input color values, wherein: the synthesized substantially rectangular spectral representation is adapted to produce the input color values if the synthesized substantially rectangular spectral representation were applied to the analysis functions corresponding to the input color values, and the first wavelength comprises a wavelength where the substantially rectangular spectral representation transitions from a lower value to the scale factor and the second wavelength comprises a wavelength where the substantially rectangular spectral representation transitions from the scale factor to the lower value.

Tristimulus-based systems and spectral or multi-spectral systems may be largely incompatible and practiced by separate enterprises. The present disclosure bridges that gap between the tristimulus-based systems and spectral or multi-spectral systems and describes methods for transforming from a tristimulus domain into a spectral domain. These transformations enable application of spectral and multi-spectral image processing methods to tristimulus image data.

As used herein, the term “image capture device” may refer to any device adapted to form an image. The image capture device captures visual information in the form of still or moving pictures. Image information (e.g., image size, image resolution, file format, and so forth) associated with such images may also be stored. Processing of stored information may also be performed. Such image capture devices may include cameras and/or line-scan cameras, flatbed scanners, and other such devices.

As used herein, the term “synthesis” may refer to generation of a signal based on the combination of entities that comprise parameters and/or functions. According to the present disclosure, synthesis of a spectral representation based on color values or image capture device sensor outputs is provided.

As used herein, the terms “actual color” and “correct color” are used interchangeably and are defined herein to mean color perceived by the human visual system.

As used herein, the term “module” may refer to a unit configured to perform certain functions. Modules may be implemented in hardware, software, firmware or combination thereof.

Section 1

FIG. 1A depicts an image processing method in accordance with an embodiment of the present disclosure. FIG. 1B depicts the expansion module of FIG. 1A, comprising a spectral synthesis module (600A) followed by a filter bank (700). As depicted in FIG. 1C, processing may not necessarily occur within an image capture device. In such case, there may only be expansion into a substantially rectangular representation followed by projection into an original domain in order to present accurate colors (explained further in section 2).

Referring now to FIG. 1A, a tristimulus representation of color values, for example, RGB (red, green, blue), resulting from an input spectrum are first expanded to a six-basis color representation RYGCBV (red, yellow, green, cyan, blue, violet). This choice for the number of primaries is partially reinforced by statistical analysis of a large database of real-world object reflectances. In this case, it is shown that 99% of the statistical variation of the reflectance dataset can be described by as few as 6 basis vectors (see reference [3], incorporated herein by reference in its entirety). The image can then be processed in the RYGCBV domain. Following processing, the RYGCBV representation of the processed image can then be projected back onto the original RGB domain. It should be noted that, although the RGB and RYGCBV color spaces are considered in the present discussion, color spaces such as YUV, YCbCr, HSV, CMYK and other color spaces known by a person skilled in the art can also be considered.

By way of example, and not limitation, the input (102A) and output (118A) may also be XYZ tristimulus values. By way of example, and not limitation, the expanded representations (105A, 115A) may comprise a number of values other than 6, such as 31 if the visible spectrum is considered as ranging from 400 nm to 700 nm in 10 nm increments. Other possibilities include using a 7 color representation (ROYGCBV) or a 10 color representation. A 6 color representation is useful because it provides a balance between accuracy and computational complexity.

Expanding RGB or XYZ values may not result in a unique spectral expansion because of metamerism (a phenomenon where two different input spectra can result in the same RGB color values). However, as long as a chosen spectral representation, if applied to analysis functions (discussed in greater detail below) corresponding to RGB or XYZ values, would result in the RGB or XYZ values resulting from the input spectrum, accurate color representation relative to actual color can be preserved.

Any given color is a spectrum of light. Such spectrum may be approximately represented according to the equation given below:

${{S\lbrack\lambda\rbrack} \cong {\hat{S}\lbrack\lambda\rbrack}} = {\sum\limits_{i}^{N}\; {C_{i}{B_{i}\lbrack\lambda\rbrack}}}$

wherein S[λ] represents an input spectrum, Ŝ[λ] in represents an approximate representation, C_(i) represents an i-th color output value, B_(i)[λ] represents an i-th basis function, and N represents the number of basis functions. For example, an approximate RGB representation can be expressed by the equation

Ŝ[λ]=C _(R) B _(R) [λ]+C _(G) B _(G) [λ]+C _(B) B _(B)[λ].

The basis functions are generally defined functions. For the CIE (International Commission on Illumination) 1931 color space (a mathematically defined color space created by the CIE in 1931; see reference [4], incorporated herein by reference in its entirety), the basis functions are narrowband with peaks at 435.8, 546.1, and 700 nanometers. For displays, the basis functions are the spectral emissions.

Basis functions are associated with matching analysis functions A_(i)[λ], which can be used to determine color output values C_(i) according to the following equation:

C_(i) = S[λ] ⋅ A_(i)[λ] = ∫_(λ = 360)^(λ = 810)S[λ]A_(i)[λ] λ

where the matching analysis functions and basis functions are related according to the equation:

A _(i) [λ]·B _(j)[λ]=δ_(ij)

and the limits of integration at 360 nm and 810 nm represent lower (λ_(min)) and upper (λ_(max)) limits of wavelengths of visible light.

The preceding equations can also be generalized for other analysis functions (e.g. can be expanded to include infrared and/or ultraviolet). While the equations above indicate orthogonal basis functions, other basis functions can be used as well. By way of example and not of limitation, basis functions could be orthogonal with respect to a matrix. Although generally not likely, it should be noted that an analysis function can be identical to its corresponding basis function. Analysis functions have meaning as well. For CIE 1931 color space, they are spectral matching functions. Analysis functions can also be spectral sensitivities of an image capture device or eye spectral sensitivities.

An embodiment of the present disclosure utilizes substantially rectangular spectra similar to those proposed by MacAdam to generate a new representation of a given color using, for instance, six basis colors RYGCBV.

MacAdam formalized a spectral representation for “maximum efficiency” reflectance spectra (see reference [5], incorporated herein by reference in its entirety). These spectra have the property that, for any desired hue and saturation, the efficiency (i.e., reflected luminance) is maximized. Such spectra can be interpreted as “optimal ink”. This family of spectra is complete any possible chromaticity can be represented. MacAdam characterized these reflectance spectra as having binary values, 0 and 1, and two transition wavelengths, λ_(↑) for a 0→1 transition and λ_(↓) for a 1→0 transition. This gives rise to band-pass and band-gap spectra, which occupy the (x, y) chromaticity domain as depicted in FIG. 2.

Although MacAdam viewed these rectangular spectra as pure reflectance spectra, it is possible to extend them for a general spectral representation of light by introducing a scale factor I:

${S_{rect}\left\lbrack {{\lambda;\lambda_{\uparrow}},\lambda_{\downarrow},I} \right\rbrack} = \left\{ {\begin{matrix} {I,{{{if}\mspace{14mu} \lambda_{\uparrow}} \leq {\lambda_{\downarrow}\mspace{14mu} {and}\mspace{14mu} \left( {\lambda \geq {\lambda_{\uparrow}\mspace{14mu} {and}\mspace{14mu} \lambda} \leq \lambda_{\downarrow}} \right)}}} \\ {I,{{{if}\mspace{14mu} \lambda_{\uparrow}} > {\lambda_{\downarrow}\mspace{14mu} {and}\mspace{14mu} \left( {\lambda \geq {\lambda_{\uparrow}\mspace{14mu} {or}\mspace{14mu} \lambda} \leq \lambda_{\downarrow}} \right)}}} \\ {0,{otherwise}} \end{matrix}.} \right.$

The three-parameter rectangular spectrum is adequate for representing all possible perceivable colors. However, real objects generally cannot be represented completely by rectangular spectra. Real objects tend to reflect or transmit some light at all wavelengths, even though reflectance may be dominant over a more limited range of wavelengths. This can be largely accounted for by adding an additional parameter, which represents a low value for the rectangular spectrum. This can be written as a four-parameter rectangular spectrum:

${S_{4,{rect}}\left\lbrack {{\lambda;\lambda_{\uparrow}},\lambda_{\downarrow},I_{h},I_{l}} \right\rbrack} = \left\{ {\begin{matrix} {I_{h},{{{if}\mspace{14mu} \lambda_{\uparrow}} \leq {\lambda_{\downarrow}\mspace{14mu} {and}\mspace{14mu} \left( {\lambda \geq {\lambda_{\uparrow}\mspace{14mu} {and}\mspace{14mu} \lambda} \leq \lambda_{\downarrow}} \right)}}} \\ {I_{h},{{{if}\mspace{14mu} \lambda_{\uparrow}} > {\lambda_{\downarrow}\mspace{14mu} {and}\mspace{14mu} \left( {\lambda \geq {\lambda_{\uparrow}\mspace{14mu} {or}\mspace{14mu} \lambda} \leq \lambda_{\downarrow}} \right)}}} \\ {I_{l},{otherwise}} \end{matrix}.} \right.$

It should be noted that the three-parameter rectangular spectrum can be represented as a four-parameter rectangular spectrum with the low value I_(l) set to 0. Spectral diagrams for both the three-parameter and four-parameter rectangular spectral representations are depicted in FIG. 3. Further discussion of the properties of rectangular spectra is given in Table A, which forms an integral part of the present disclosure.

A given color represented by (x, y) coordinates in chromaticity space can be represented as a rectangular bandpass or bandgap spectrum (see FIG. 4). Alternatively, the [λ_(min), λ_(max)] domain (the wavelengths of visible light) itself can be interpreted as circular, allowing all spectra to have a band-pass form. In such a case, what were formerly band-gap spectra become band-pass spectra that cross the λ_(min)/λ_(max) point (see FIGS. 5A-5C). The highlighted portions of the circular domains depicted in FIGS. 5B and 5C indicate wavelengths at which the value of the rectangular spectrum is equal to the scale factor I. Interpretation of the [λ_(min), λ_(max)] domain as circular resulting in all spectra having a band-pass form is also depicted in FIG. 18, explained further in Table A.

In order to perform spectral expansion on a set of original tristimulus values (which can be from any color space) from an image to be processed, chromaticity coordinates are first used to compute λ_(↑) and λ_(↓). The scale factor, I, is then derived from λ_(↑), λ_(↓) and the original tristimulus values.

FIG. 6A depicts a spectral synthesis module (600A) that can be utilized in the image processing method depicted in FIG. 1A (specifically, used as part of the expansion depicted in FIG. 1B). XYZ tristimulus values (610A) are input into a wavelength determination module (620A, depicted in detail in FIG. 6B) to determine the transition wavelengths (615B) λ_(↑) and λ_(↓). The computations can involve computing (600B) values x and y (605B) based on the XYZ tristimulus values (610A):

$x = \frac{X}{X + Y + Z}$ $y = {\frac{Y}{X + Y + Z}.}$

A two-dimensional lookup table (2D-LUT, 610B) can then be utilized to map the values x and y to determine the transition wavelengths (615B) λ_(↑) and λ_(↓). Based on these transition wavelengths λ_(↑) and λ_(↓), the scale factor I can be determined by a scale factor computation module (650A) comprised of a circular integration module (640A) and a division module (660A). The circular integration module (640A) can perform circular integration of any one of the spectral analysis functions [ x[λ], y[λ], z[λ]] (630A) over an interval defined by the transition wavelengths λ_(↑) and λ_(↓). By way of example, and not limitation, FIG. 6A depicts circular integration of x[λ]. A person skilled in the art can understand that y[λ] or z[λ] any combination of the spectral analysis functions can be used for performing circular integration as well. A result of the circular integration module (640A) is then provided to the division module (660A) to produce a scale factor I. Specifically, the scale factor I can be given by the following equation:

$I = {\frac{X}{\int_{\uparrow {, \downarrow}}{{\overset{\_}{x}\lbrack\lambda\rbrack}\ {\lbrack\lambda\rbrack}}} = {\frac{Y}{\int_{\uparrow {, \downarrow}}{{\overset{\_}{y}\lbrack\lambda\rbrack}{\lambda}}} = {\frac{Z}{\int_{\uparrow {, \downarrow}}{{\overset{\_}{z}\lbrack\lambda\rbrack}{\lambda}}}.}}}$

These integrals can be computed, for instance, with two table look-ups and addition/subtraction operations. In this manner, the three parameters of the rectangular spectrum [λ_(↑), λ_(↓), I] (670) are determined to synthesize the rectangular spectrum.

As depicted in FIG. 7, a synthesized rectangular spectrum can be filtered (the filter banks can but need not be equal in bandwidth) to produce RYGCBV color values. At this point, image processing may be performed on the RYGCBV color values.

According to several embodiments of the present disclosure, a class of processing operations applicable to both tristimulus and spectral or multi-spectral representations is described. For traditional RGB data associated with an image, there are two common mathematical operations: multiplication by a 3×3 matrix and independent non-linear transformation of the individual RGB channels. Matrix multiplication is commonly used to adjust color and/or the effects of the illuminant. Non-linear transformation is often referred to as tone-mapping because the non-linear transformation alters visual tone (brightness and contrast) of the image. Note that any number of 3×3 matrix multiplications can be collapsed into a single 3×3 matrix multiplication and that similarly any number of one dimensional (1D) transformations of each of the RGB channels can similarly be collapsed into just one non-linear transformation for each of those three channels. Thus, an image can be processed through a set of three non-linear transformations, one for each channel associated with the image, followed by a matrix multiplication.

For example, even with only the traditional three RGB channels, a transformation as specified above comprising performing non-linear transformation followed by matrix multiplication can accurately encapsulate the differences between the image formats common today, which possess limited intensity range and color, to potential future formats possessing larger intensity range (often called dynamic range) and color gamut.

This class of transformations can be generalized to a multi-spectral case for any number of color channels through the following relation:

O _(i) =M _(ij) T _(j) [I _(j)]

i,j=1, . . . ,N

where I_(j) denotes a j-th input color channel value (e.g., R, G, or B in an RGB representation or R, Y, G, C, B, or V, in an RYGCBV representation), T_(j) denotes non-linear transformation applied to I_(j), M_(ij) denotes an N×N matrix, and O_(i) denotes an i-th output color channel value.

The spectral image processing method discussed above can be applied to spectral color correction. In practice, primary color correction can be applied through modifications directly to the RGB channels independently. While these modifications can account for primary manipulations that will need to be performed, it is difficult to manipulate specific hues only. For example, it is difficult to make a yellow hue more intense without modifying R and G (the adjacent colors) or B (reducing blue has the effect of directly increasing yellow). Fundamentally, this is because three color channels are sufficient to match a given color, but insufficient for hue control as there are four fundamental hues perceptually, red vs. green and blue vs. yellow, as described by opponent color theory.

In practice, this can be handled by secondary color correction. Secondary color correction transforms the RGB data into an HSL (hue-saturation-luminance) representation and modifies HSL values conditionally over a specified range of hue-saturation-luminance values. Since yellow is half-way between red and green, cyan is half-way between green and blue, and magenta is half-way between blue and red, secondary color correction is often implemented with 6 hues, RYGCBM, which can be referred to as 6-axis secondary color correction. Primary and global secondary color correction can be integrated in the methods of spectral image processing in accordance with the present disclosure utilizing, for example, RYGCBV color values (note that M is not necessary as it is always a BR mixture, unlike C and Y). A sigmoid tone curve, depicted in FIG. 8, can be modified for adjustment of amplitude, gamma, toe, and shoulder. Re-saturation can be performed by adding white or by spectral broadening.

Section 2

According to additional embodiments of the present disclosure, spectral synthesis methods can be applied to image capture device processing. FIG. 9 depicts a processing method performed in modern digital image capture devices (see reference [6], incorporated herein by reference in its entirety). Raw image capture device sensor outputs (905) (e.g., R, G1, G2, and B values) resulting from an input spectrum S(λ) are first processed (910) according to a method comprising one or more of linearization (if necessary), deBayering (e.g., for single-chip sensors), and white-balancing. A Bayer filter mosaic is a color filter array for arranging RGB color filters on a square grid of photosensors in an image capture device. Obtaining a full-color image involves reversing this process in a step known as deBayering (see reference [7], incorporated herein by reference in its entirety). White-balancing is an adjustment of intensities of primary colors (e.g., red, green, and blue) to correctly render specific colors (see reference [8], incorporated herein by reference in its entirety).

Data obtained from such processing (910) of the raw image capture device sensor outputs (905) are then generally transformed to an output color space (e.g., RGB color space in FIG. 9) by a 3×3 matrix (915). The transformation output can be tone processed and/or clipped (920) to a specific range. The processing performed in each of blocks (910), (915), and (920) yields an output color space (925) from the raw image capture device sensor outputs (905).

According to several embodiments of the present disclosure, image capture device processing methods alternative to the 3×3 matrix are presented. The 3×3 matrix alone may not be sufficient to describe accurate transformation of image capture device sensor outputs to output colors (925).

In order to focus on the possible issues with this 3×3 matrix (915), consider the simplified configuration depicted in FIG. 10.

For the present discussion, consider an image capture device comprising a red (R), green (G), and blue (B) channels (e.g., RGB is considered an input color space) and consider CIE [X, Y, Z] as an output color space (1020). It should be noted that, although the RGB and CIE color spaces are considered in the present discussion, color spaces such as YUV, YCbCr, HSV, CMYK and other color spaces known by a person skilled in the art can also be considered.

Tristimulus values [X, Y, Z] can be determined from an input spectrum S[λ] and color matching functions [ x[λ], y[λ]], z[λ] defined over an interval [λ_(min), λ_(max)] by the equations

X=∫ _(λ) _(min) ^(λ) ^(max) x[λ]S[λ]dλ

Y=∫ _(λ) _(min) ^(λ) ^(max) y[λ]S[λ]dλ

Z=∫ _(λ) _(min) ^(λ) ^(max) z[λ]I[λ]dλ

where the interval [λ_(min), λ_(max)] encompasses wavelengths of light generally perceptible by a human visual system. FIG. 12 depicts an example of such functions as determined by CIE in 1931.

Similarly, image capture device sensor outputs [R_(S), G_(S), B_(S)] (1010) (where the subscript indicates sensor) are determined by image capture device spectral sensitivities [ r _(S)[λ], g _(S)[λ], b _(S)[λ]] using analogous equations

R _(S)=∫_(λ) _(min) ^(λ) ^(max) r _(S) [λ]S[λ]dλ

G _(S)=∫_(λ) _(min) ^(λ) ^(max) g _(S) [λ]S[λ]dλ

B _(S)=∫_(λ) _(min) ^(λ) ^(max) b _(S) [λ]S[λ]dλ

where the image capture device spectral sensitivities represent wavelength response of image capture device color channels. FIG. 13 depicts an example of such functions for an example modern digital camera.

FIG. 11 depicts that an output color [X_(S), Y_(S), Z_(S)] (1135) from the image capture device, obtained through applying a 3×3 matrix transformation (1130) to image capture device sensor outputs [R_(S), G_(S), B_(S)] (1125), may not be the same as actual color [X, Y, Z] (1105) perceived by the human visual system. Color accuracy between the output color from the image capture device [X_(S), Y_(S), Z_(S)] (1135) and the actual color [X, Y, Z] (1105) can generally be guaranteed only when the image capture device spectral sensitivities [ r _(S)[λ], g _(S)[λ], b _(S)[λ]] (1120) are an invertible linear combination of color matching functions (1110). Mathematically, this occurs when there exists a 3×3 matrix Q such that

$\begin{bmatrix} {{\overset{\_}{r}}_{s}\lbrack\lambda\rbrack} \\ {{\overset{\_}{g}}_{s}\lbrack\lambda\rbrack} \\ {{\overset{\_}{b}}_{s}\lbrack\lambda\rbrack} \end{bmatrix} = {Q\begin{bmatrix} {\overset{\_}{x}\lbrack\lambda\rbrack} \\ {\overset{\_}{y}\lbrack\lambda\rbrack} \\ {\overset{\_}{z}\lbrack\lambda\rbrack} \end{bmatrix}}$

where Q⁻¹ exists.

Multiplying the above equation on both sides by the input spectrum S[λ] (1115) and integrating both sides yield the result

$\begin{bmatrix} R_{S} \\ G_{S} \\ B_{S} \end{bmatrix} = {{Q\begin{bmatrix} X \\ Y \\ Z \end{bmatrix}}.}$

It follows that

$\begin{bmatrix} X \\ Y \\ Z \end{bmatrix} = {{Q^{- 1}\begin{bmatrix} R_{S} \\ G_{S} \\ B_{S} \end{bmatrix}}.}$

With reference back to FIG. 11, to obtain the actual color [X, Y, Z] (1105) directly, ideally the image capture device spectral sensitivities [ r _(S)[λ], g _(S)[λ], b _(S)[λ]] (1120) should be identical to the color matching functions [ x[λ], y[λ], z[λ]] (1110). If these ideal spectral sensitivities (which equal the color matching functions) are modified by a matrix Q to produce the actual image capture device spectral sensitivities, then such modification can be undone (e.g., by transforming the image capture device sensor outputs (1125) with Q⁻¹) to produce the actual color [X, Y, Z] (1105).

If the relationship between the image capture device spectral sensitivities [ r _(S)[λ], g _(S)[λ], b _(S)[λ]] (1120) and the color matching functions [ x[λ], y[λ], z[λ]] (1110) is more complex than a linear transform, a 3×3 matrix may not be sufficient.

Differences between the color matching functions and image capture device spectral sensitivities can be significant. FIG. 12 depicts the CIE (International Commission on Illumination) 1931 color matching functions while FIG. 13 depicts exemplary spectral sensitivities for a modern digital motion picture camera (the ARRI D-21). In 1931, the CIE created a mathematically defined color space known as CIE 1931. The color matching functions of FIG. 12 correspond to this color space (see reference [4], incorporated herein by reference in its entirety). There is no linear combination of color matching functions that would yield the spectral sensitivities for this camera. Specifically, the CIE 1931 color matching functions are smooth, whereas the camera spectral sensitivities are not. Consideration of such camera response, as provided by the camera spectral sensitivities, is generally not taken into account in the sensor to output processing.

Although applying a matrix transform may not be sufficient for transforming the image capture device sensor outputs to the actual color, accurate processing from the image capture device sensor outputs to an output color space can exhibit overall linearity. That is, if {right arrow over (C)}_(S) represents the sensor output [R_(S), G_(S), B_(S)], {right arrow over (C)}_(out) represents the output color [X_(S), Y_(S), Z_(S)], and P[ ] represents processing between the image capture device sensor output and the output color, e.g.,

{right arrow over (C)} _(out) =P[{right arrow over (C)} _(S)]

then multiplying the input {right arrow over (C)}_(S) by some constant α causes outputs to change by the same factor,

P[α{right arrow over (C)} _(S) ]=α{right arrow over (C)} _(out).

However, even when such properties of linearity are exhibited, the processing may not necessarily be by means of applying a matrix exclusively. Such a matrix can be determined by minimizing a particular error metric over a training set of color stimuli. Some recommended procedures for determining a matrix in this manner are described in reference [6], incorporated by reference herein in its entirety.

According to several embodiments of the present disclosure, methods and systems for generating actual color perceived by a human visual system from given image capture device spectral sensitivities are described.

FIG. 14A depicts an embodiment of a method and system for synthesizing a spectral representation Ŝ[λ] (1435A) that, if applied to the image capture device, can produce the observed image capture device sensor outputs [R_(S), G_(S), B_(S)] (1425A). This method can be utilized with negative tristimulus values.

The synthesized spectrum Ŝ[λ] (1435A) can produce the observed image capture device sensor outputs [R_(S), G_(S), B_(S)] (1425A) regardless of whether or not the synthesized spectrum Ŝ[λ] (1435A) is an actual spectrum S[λ] (1415A). Specifically, the synthesized spectrum Ŝ[λ] (1435A) can produce the correct [X, Y, Z] (1405A, 1445A).

Once the synthesized spectrum Ŝ[λ] (1435A) has been determined, the correct output color [X, Y, Z] (1405A, 1445A) can be obtained.

According to several embodiments of the present disclosure, substantially rectangular spectra can be utilized as the synthesized spectrum Ŝ[λ](1435A). As used in the present disclosure, the term “substantially rectangular” may refer to a shape of a spectrum that may closely approximate a rectangular shape, but is not necessarily exactly rectangular in shape. By way of example and not of limitation, a spectrum characterized by side walls that are not exactly perpendicular to the horizontal (λ) axis can be considered substantially rectangular. By way of further example and not of limitation, a spectrum characterized by a small range of maximum values rather than only one maximum value can also be considered substantially rectangular. Substantially rectangular spectra can be continuous functions or discrete wavelength (sampled) functions. Examples of continuous substantially rectangular spectra and discrete wavelength substantially rectangular spectra are depicted in FIGS. 14B and 14C, respectively.

FIGS. 15A-15D depict an image capture device processing method, in accordance with an embodiment of the present disclosure, using synthesized rectangular spectra.

FIG. 15A depicts an actual image spectrum S[λ] (1505A) being captured by an image capture device. The actual image spectrum S[λ] (1505A) is applied to the image capture device spectral sensitivities [ r _(S)[λ], g _(S)[λ], b _(S)[λ]] (1510A) to produce image capture device sensor outputs [R_(S), G_(S), B_(S)] (1515A). The image capture device sensor outputs [R_(S), G_(S), B_(S)] (1515A) can be input to a spectral synthesis module (1520A) to generate a spectral representation characterized by rectangular spectral parameters [λ_(↑), λ_(↓), I] (1525A). The synthesized rectangular spectrum (1525A) is adapted to produce the observed image capture device sensor outputs [R_(S), G_(S), B_(S)] (1515A) if applied to the image capture device:

R _(S) =I∫ _(λ) _(↑) _(,λ) _(↓) r _(S) [λ]dλ

G _(S) =I∫ _(λ) _(↑) _(,λ) _(↓) g _(S) [λ]dλ

B _(S) =I∫ _(λ) _(↑) _(,λ) _(↓) b _(S) [λ]dλ

where the integration symbol denotes circular integration over the λ domain

$\int_{\lambda_{\uparrow},\lambda_{\downarrow}}{\equiv \ \begin{matrix} {{\int_{\lambda_{\uparrow}}^{\lambda_{\downarrow}},}\ } & {{{if}\mspace{14mu} \lambda_{\uparrow}} \leq \lambda_{\downarrow}} \\ {{\int_{\lambda_{\uparrow}}^{\lambda_{\max}}{+ \int_{\lambda_{\min}}^{\lambda_{\downarrow}}}},} & {{otherwise}.} \end{matrix}}$

The synthesized rectangular spectrum (1525A) can be applied to a spectral application module (1530A) to produce correct color outputs [X, Y, Z] (1535A).

Alternatively, the [λ_(min), λ_(max)] domain itself can be interpreted as circular, allowing all spectra to have a band-pass form. In such a case, what were formerly band-gap spectra become band-pass spectra that cross the λ_(min)/λ_(max) point.

Hence, mathematically, the first step is to solve for [λ_(↑), λ_(↓), I] (1525A) given the image capture device sensor outputs [R_(S), G_(S), B_(S)] (1515A). Once parameters of the rectangular spectrum have been determined, computation of the correct output [X, Y, Z] (1535A) can follow directly:

X=I∫ _(λ) _(↑) _(,λ) _(↓) x[λ]dλ

Y=I∫ _(λ) _(↑) _(,λ) _(↓) y[λ]dλ

Z=I∫ _(λ) _(↑) _(,λ) _(↓) z[λ]dλ

This process is depicted in FIG. 15D. Specifically, the wavelengths λ_(↑) and λ_(↓) in addition to the color matching functions [ x[λ], y[λ], z[λ]] are sent to a circular integration module (1505D). Results of the integrations are individually multiplied by the scale factor I via a multiplication module (1510D) to generate correct color output values [X, Y, Z] (1535A).

Performance of this process of using a rectangular spectral representation Ŝ[λ] equivalent to the original image's spectral representation S[λ] to obtain correct color output for image capture devices can take any of several forms, each with different complexity and performance.

In one embodiment, with reference back to FIG. 15A, a generally computation-intensive implementation can first determine rectangular parameters [λ_(↑), λ_(↓), I] (1525A) and then produce the output color [X, Y, Z] (1535A) from rectangular spectra characterized by [λ_(↑), λ_(↓), I] (1525A). Because the number of degrees of freedom (e.g., three in this case) is conserved, this process provides a mapping from [R_(S), G_(S), B_(S)] to [X, Y, Z] as depicted in FIG. 16.

A three-dimensional look up table (3D-LUT) can be utilized to perform this mapping. It is also possible to perform computations (to be provided below) and then use a 2D-LUT based on the computations to determine the transition wavelengths λ_(↑) and λ_(↓). The transition wavelengths can be then used to determine the scale factor I. This process is depicted in FIG. 15B.

FIG. 15B depicts a spectral synthesis module (1520A) that can be utilized in the image capture device processing method depicted in FIG. 15A. Image capture device sensor outputs [R_(S), G_(S), B_(S)] (1515A) are input into a wavelength determination module (1505B, depicted in further detail in FIG. 15C) to determine λ_(↑) and λ_(↓). The computations can involve computing (1500C) values p and q (1505C) based on the image capture device sensor outputs [R_(S), G_(S), B_(S)] (1515A):

$p = \frac{R_{S}}{R_{S} + G_{S} + B_{S}}$ $q = {\frac{G_{S}}{R_{S} + G_{S} + B_{S}}.}$

A two-dimensional lookup table (2D-LUT, 1510C) can then be utilized to map the values p and q (1505C) to determine the transition wavelengths λ_(↑) and λ_(↓) (1515C). Based on these transition wavelengths λ_(↑) and λ_(↓), the scale factor I can be determined by a scale factor computation module (1520B) comprised of a circular integration module (1510B) and a division module (1515B). The circular integration module (1510B) can perform circular integration of any one of the image capture device spectral sensitivities [ r _(S)[λ], g _(S)[λ], b _(S)[λ]] (1510A) over an interval defined by the transition wavelengths λ_(↑) and λ_(↓). A result of the circular integration module (1510B) is then provided to the division module (1515B) to produce a scale factor I. Specifically, scale factor I can be given by the following equation:

$I = {\frac{R_{S}}{\int_{\uparrow {, \downarrow}}{{{\overset{\_}{r}}_{s}\lbrack\lambda\rbrack}{\lambda}}} = {\frac{G_{s}}{\int_{\uparrow {, \downarrow}}{{{\overset{\_}{g}}_{s}\lbrack\lambda\rbrack}{\lambda}}} = {\frac{B_{S}}{\int_{\uparrow {, \downarrow}}{{{\overset{\_}{b}}_{s}\lbrack\lambda\rbrack}{\lambda}}}.}}}$

These integrals can be computed, for instance, with two table look-ups and addition/subtraction operations.

The method of the preceding discussion involving synthesis of a rectangular spectral representation equivalent to the spectral representation of the image captured by the image capture device can be utilized in examining and characterizing image capture device accuracy. With some spectra, this method can result in an output rectangular spectrum which is identical to the input rectangular spectrum for some set of stimuli.

According to several embodiments of the present disclosure, a method for determining this set of stimuli is provided as follows. The method can first comprise simulating exposure of a cube in [λ_(↑), λ_(↓), I] (1705) to the image capture device's spectral sensitivities [ r _(S)[λ], g _(S)[λ], b _(S)[λ]] (1710) to produce simulated image capture device sensor outputs [R_(S), G_(S), B_(S)] (1715). The cube (1705) provides a representation of rectangular spectra in rectangular space defined by three dimensions [λ_(↑), λ_(↓), I]. Specifically, each point in the cube corresponds to a rectangular spectrum, where the rectangular spectrum is characterized by a scale factor I, a first wavelength λ_(↑) being a wavelength where the input spectrum transitions from zero to the scale factor, and a second wavelength λ_(↓) being a wavelength where the input spectrum transitions from the input scale factor to zero. A next step is to determine by comparison (1730) a set of spectra for which an output [{tilde over (λ)}_(↑), {tilde over (λ)}_(↓), Ĩ] (1725) of the previously described spectral synthesis method is the same as the input [λ_(↑), λ_(↓), I] (1705) as depicted in FIG. 17. The set of rectangular spectra which can be recovered exactly using this method constitutes an image capture device gamut. Rectangular spectra which cannot be recovered exactly constitute what is commonly referred to as image capture device metamerism.

The examples set forth above are provided to give those of ordinary skill in the art a complete disclosure and description of how to make and use the embodiments of the spectral image processing of the disclosure, and are not intended to limit the scope of what the inventor/inventors regard as their disclosure.

Modifications of the above-described modes for carrying out the methods and systems herein disclosed that are obvious to persons of skill in the art are intended to be within the scope of the following claims. All patents and publications mentioned in the specification are indicative of the levels of skill of those skilled in the art to which the disclosure pertains. All references cited in this disclosure are incorporated by reference to the same extent as if each reference had been incorporated by reference in its entirety individually.

It is to be understood that the disclosure is not limited to particular methods or systems, which can, of course, vary. It is also to be understood that the terminology used herein is for the purpose of describing particular embodiments only, and is not intended to be limiting. As used in this specification and the appended claims, the singular forms “a”, “an”, and “the” include plural referents unless the content clearly dictates otherwise. The term “plurality” includes two or more referents unless the content clearly dictates otherwise. Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which the disclosure pertains.

The methods and systems described in the present disclosure may be implemented in hardware, software, firmware or combination thereof. Features described as blocks, modules or components may be implemented together (e.g., in a logic device such as an integrated logic device) or separately (e.g., as separate connected logic devices). The software portion of the methods of the present disclosure may comprise a computer-readable medium which comprises instructions that, when executed, perform, at least in part, the described methods. The computer-readable medium may comprise, for example, a random access memory (RAM) and/or a read-only memory (ROM). The instructions may be executed by a processor (e.g., a digital signal processor (DSP), an application specific integrated circuit (ASIC), or a field programmable gate array (FPGA)).

A number of embodiments of the disclosure have been described. Nevertheless, it will be understood that various modifications can be made without departing from the spirit and scope of the present disclosure. Accordingly, other embodiments are within the scope of the following claims.

LIST OF REFERENCES

-   [1] wikipedia.org/wiki/Multispectral_image, retrieved 6 Dec. 2011 -   [2] Wyszecki, G. and Stiles, W. S., Color Science: Concepts and     Methods, Quantitative Data and Formulae, Wiley-Interscience, 2002,     pp, 187-188. -   [3] M. Parmar, et al, “A Database of High Dynamic Range Visible and     Near-infrared Multispectral Images”, Proc. IS&T/SPIE Electronic     Imaging 2008: Digital Photography IV. -   [4] wikipedia.org/wiki/CIE_(—)1931_color_space, retrieved 29 Nov.     2011. -   [5] MacAdam, David L., “The Theory of the Maximum Visual Efficiency     of Color Materials,” J. Optical Soc. Am., Vol. 25, pp. 249-252,     1935. -   [6] “Color Image Processing Pipeline. A General Survey of Digital     Still Camera Processing”, IEEE Signal Processing Magazine, vol. 22,     no. 1, pp. 34-43, 2005. -   [7] wikipedia.org/wiki/Bayer_filter, retrieved 6 Dec. 2011 -   [8] wikipedia.org/wiki/White_balance, retrieved 6 Dec. 2011 -   [9] Logvinenko, Alexander D., “An Object Color Space”, J. Vision.     9(11):5, pp. 1-23, 2009

TABLE A Rectangular Spectra Rectangular spectra possess many useful properties: a) MacAdam efficiency b) Complete (all possible chromaticity) c) 3 degrees of freedom d) Maximally compact spectra

The last property, compactness, arises because there are fundamentally two ways to de-saturate any chromaticity: broaden the spectrum or add white. Logvinenko (see reference [9], incorporated herein by reference in its entirety) exploited this property to produce a more general form for the reflective spectra of objects (and also replaced λ_(↑) and λ_(↓) with an equivalent form using the center wavelength and a signed bandwidth where negative bandwidths were used to represent bandgap spectra).

FIG. 2 depicts how the band-pass and band-gap behavior appears in chromaticity space. Note that the boundary between band-pass and band-gap regions has two segments: one segment runs from the equal-energy white point to the maximum wavelength red, and the other segment runs from the white point to the minimum wavelength blue. These are referred to as the white-red boundary and the white-blue boundary, respectively.

It is possible to use circular integrals to hide the distinction between band-pass and band-gap spectra. This can also be achieved by considering the wavelength domain to be circular, also eliminating the distinction between band-pass and band-gap spectra. FIGS. 5A-5C depict this circular domain over which the integrals are evaluated. Specifically, FIG. 5A depicts the circular wavelength domain, FIG. 5B depicts a band-pass rectangular spectrum, and FIG. 5C depicts a band-gap rectangular spectrum.

Note that λ_(min) and λ_(max) are the same point in the circular representation and that increases in the counter-clockwise direction. In the circular representation the integral is always from λ_(↑) to λ_(↓) and the solution is always band-pass as depicted in FIG. 18. An alternative visualization of this would be to draw these spectra on the outside of a tube so that λ_(min) and λ_(max) can be the same point.

Referring now to FIG. 19, the white-red and white-blue boundary segments of FIG. 2 are where λ_(↑) or λ_(↓) fall on the λ_(min)/λ_(max) boundary. The white-red boundary is where λ_(↓) falls on λ_(max), and the white-blue boundary is where λ_(↑) falls on λ_(min).

It is illustrative to see how the (x,y) solution space for band-pass and band-gap spectra depicted in FIG. 2 maps into the domain in [λ_(↑), λ_(↓)] space depicted in FIGS. 20A and 20B.

An equal energy white point E is where λ_(↑)=λ_(min) and λ_(↓)=λ_(max). The diagonal line is on the spectral local if approached from the band-pass side or near the white-point if approached from the band-gap side. The upper triangular region for band-pass spectra is a closed set (it includes its boundary) while the triangular region for band-gap spectra is open (it does not include its boundary). FIG. 15B depicts the trajectory in the two-dimensional λ-space of relatively narrow-band spectra of bandwidth δ and a center frequency that traverses the circular lambda domain. As the center frequency goes from λ_(min)+δ/2 to λ_(max)−δ/2, the hues pass through violet (V), blue (B), cyan (C), green (G), yellow (Y) and red (R). When the center frequency is within δ/2 of [λ_(↑), λ_(↓)] the hue passes through magenta. 

1-35. (canceled)
 36. A method for synthesizing a substantially rectangular spectral representation of input color values in a tristimulus representation, the input color values resulting from an input spectrum of an image, wherein each of the input color values corresponds to a weight of a corresponding basis function of the tristimulus representation, wherein the basis functions of the tristimulus representation are defined functions, the method comprising: providing the input color values, wherein each input color value is associated with a corresponding analysis function; wherein the analysis functions are associated with the basis functions of the tristimulus representation; determining a first wavelength and a second wavelength of the substantially rectangular spectral representation based on the input color values; wherein the determining of the first wavelength and the second wavelength comprises: dividing two of the input color values by a sum of all of the input color values; and using a two dimensional look-up table to map results of the dividing to determine the first wavelength and the second wavelength; and computing a scale factor based on the first wavelength, the second wavelength, and the input color values and its corresponding analysis function; wherein the computing of the scale factor comprises: performing a circular integration of the corresponding analysis function of the one of the input color values across an interval defined by the first wavelength and the second wavelength; and dividing the one of the input color values by a result of the circular integration to compute the scale factor; wherein: the synthesized substantially rectangular spectral representation is adapted to produce the input color values if the synthesized substantially rectangular spectral representation were applied to the analysis functions corresponding to the input color values, at the first wavelength the substantially rectangular spectral representation transitions from a lower value to the scale factor and at the second wavelength the substantially rectangular spectral representation transitions from the scale factor to the lower value, and if the first wavelength is smaller than the second wavelength, in between the first and second wavelengths, the substantially rectangular spectral representation takes on the scale factor, and if the first wavelength is greater than the second wavelength, in between the first and second wavelengths, the substantially rectangular spectral representation takes on the lower value; and the lower value of the rectangular spectral representation is equal to zero.
 37. The method of claim 36, wherein the analysis functions comprise spectral sensitivities of an image capture device or eye spectral sensitivities.
 38. A method of spectral expansion, the method comprising: synthesizing a substantially rectangular spectral representation according to the method of claim 36; and applying a plurality of filter banks to the synthesized substantially rectangular spectral representation to generate expanded color values in an expanded representation, wherein a number of expanded color values in the expanded representation is greater than a number of input color values in the tristimulus representation, to expand a spectrum represented by the input color values.
 39. A method of image processing, the method comprising: performing spectral expansion according to the method of claim 38; processing the expanded color values; and projecting the processed expanded color values into a color space corresponding to the input color values.
 40. The method of claim 39, wherein the processing comprises spectral color correction.
 41. The method of claim 39, wherein the processing comprises a nonlinear transformation followed by a matrix multiplication.
 42. The method of claim 36, wherein the input color values are defined for RGB (red, green, blue) color space.
 43. The method of claim 36, wherein the input color values are defined for XYZ (CIE 1931) color space.
 44. The method of claim 38, wherein the expanded color values are defined for RYGCBV (red, yellow, green, cyan, blue, violet) color space.
 45. The method of claim 38, wherein the expanded color values are defined for ROYGCBV (red, orange, yellow, green, cyan, blue, violet) color space.
 46. The method of claim 38, wherein the input color values are defined for RGB (red, green, blue) color space and the expanded color values are defined for RYGCBV (red, yellow, green, cyan, blue, violet) color space.
 47. The method of claim 38, wherein the input color values are defined for RGB (red, green, blue) color space and the expanded color values are defined for ROYGCBV (red, orange, yellow, green, cyan, blue, violet) color space.
 48. The method of claim 38, wherein the input color values are defined for XYZ (CIE 1931) color space and the expanded color values are defined for RYGCBV (red, yellow, green, cyan, blue, violet) color space.
 49. The method of claim 38, wherein the input color values are defined for XYZ (CIE 1931) color space and the expanded color values are defined for ROYGCBV (red, orange, yellow, green, cyan, blue, violet) color space.
 50. The method of claim 36, wherein the input color values may contain negative values.
 51. A system that is configured to synthesize a substantially rectangular spectral representation of input color values in a tristimulus representation, the input color values resulting from an input spectrum of an image, wherein each of the input color values corresponds to a weight of a corresponding basis function of the tristimulus representation, wherein the basis functions of the tristimulus representation are defined functions, wherein each input color value is associated with a corresponding analysis function, wherein the analysis functions are associated with the basis functions of the tristimulus representation, the system comprising: a wavelength determination module that is configured to determine a first wavelength and a second wavelength of the substantially rectangular spectral representation based on the input color values; wherein the first wavelength and the second wavelength is determined by dividing two of the input color values by a sum of all of the input color values, and by using a two dimensional look-up table to map results of the dividing to determine the first wavelength and the second wavelength; and a scale factor computation module that is configured to compute a scale factor based on the first wavelength, the second wavelength, and the input color values and its corresponding analysis function, wherein the scale factor is computed by performing a circular integration of the corresponding analysis function of the one of the input color values across an interval defined by the first wavelength and the second wavelength, and by dividing the one of the input color values by a result of the circular integration to compute the scale factor; wherein: the synthesized substantially rectangular spectral representation is adapted to produce the input color values if the synthesized substantially rectangular spectral representation were applied to the analysis functions corresponding to the input color values, and at the first wavelength the substantially rectangular spectral representation transitions from a lower value to the scale factor and at the second wavelength the substantially rectangular spectral representation transitions from the scale factor to the lower value; if the first wavelength is smaller than the second wavelength, in between the first and second wavelengths, the substantially rectangular spectral representation takes on the scale factor, and if the first wavelength is greater than the second wavelength, in between the first and second wavelengths, the substantially rectangular spectral representation takes on the lower value; the lower value of the rectangular spectral representation is equal to zero.
 52. The system of claim 51, wherein the analysis functions comprise spectral sensitivities of an image capture device or eye spectral sensitivities.
 53. A spectral expansion system, the system comprising: a spectral synthesis module comprising the system of claim 51 that is configured to synthesize a substantially rectangular spectral representation; and a plurality of filter banks that is configured generate expanded color values in an expanded representation from the synthesized substantially rectangular spectral representation, wherein a number of expanded color values in the expanded representation is greater than a number of input color values in the tristimulus representation, to expand a spectrum represented by the input color values.
 54. An image processing system, the system comprising: a spectral expansion system according to claim 53; a processing module that is configured to process the expanded color values; and a projection module that is configured to project the processed expanded color values into a color space corresponding to the input color values.
 55. The system of claim 54, wherein the processing module comprises a spectral color correction module.
 56. The system of claim 54, wherein the processing module is configured to perform a nonlinear transformation followed by a matrix multiplication.
 57. The system of claim 51, wherein the input color values are defined for RGB (red, green, blue) color space.
 58. The system of claim 51, wherein the input color values are defined for XYZ (CIE 1931) color space.
 59. The system of claim 53, wherein the expanded color values are defined for RYGCBV (red, yellow, green, cyan, blue, violet) color space.
 60. The system of claim 53, wherein the expanded color values are defined for ROYGCBV (red, orange, yellow, green, cyan, blue, violet) color space.
 61. The system of claim 53, wherein the input color values are defined for RGB (red, green, blue) color space and the expanded color values are defined for RYGCBV (red, yellow, green, cyan, blue, violet) color space.
 62. The system of claim 53, wherein the input color values are defined for RGB (red, green, blue) color space and the expanded color values are defined for ROYGCBV (red, orange, yellow, green, cyan, blue, violet) color space.
 63. The system of claim 53, wherein the input color values are defined for XYZ (CIE 1931) color space and the expanded color values are defined for RYGCBV (red, yellow, green, cyan, blue, violet) color space.
 64. The system of claim 53, wherein the input color values are defined for XYZ (CIE 1931) color space and the expanded color values are defined for ROYGCBV (red, orange, yellow, green, cyan, blue, violet) color space.
 65. The system of claim 51, wherein the input color values may contain negative values.
 66. A non-transitory computer-readable medium containing a set of instructions that causes a computer to perform the method recited in claim
 36. 